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Duality in nondifferentiable minimax fractional programming with B-(p, r)-invexity

Izhar Ahmad12*, SK Gupta3, N Kailey4 and Ravi P Agarwal15

Author Affiliations

1 Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

2 Department of Mathematics, Aligarh Muslim University, Aligarh-202 002, India

3 Department of Mathematics, Indian Institute of Technology Patna, Patna-800 013, India

4 School of Mathematics and Computer Applications, Thapar University, Patiala-147 004, India

5 Department of Mathematics, Texas A & M University - Kingsville, 700 University Blvd. Kingsville, TX 78363-8202, USA

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Journal of Inequalities and Applications 2011, 2011:75  doi:10.1186/1029-242X-2011-75

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2011/1/75


Received:6 November 2010
Accepted:30 September 2011
Published:30 September 2011

© 2011 Ahmad et al; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this article, we are concerned with a nondifferentiable minimax fractional programming problem. We derive the sufficient condition for an optimal solution to the problem and then establish weak, strong, and strict converse duality theorems for the problem and its dual problem under B-(p, r)-invexity assumptions. Examples are given to show that B-(p, r)-invex functions are generalization of (p, r)-invex and convex functions

AMS Subject Classification: 90C32; 90C46; 49J35.

Keywords:
nondifferentiable fractional programming; optimality conditions; B-(p, r)-invex function; duality theorems

1 Introduction

The mathematical programming problem in which the objective function is a ratio of two numerical functions is called a fractional programming problem. Fractional programming is used in various fields of study. Most extensively, it is used in business and economic situations, mainly in the situations of deficit of financial resources. Fractional programming problems have arisen in multiobjective programming [1,2], game theory [3], and goal programming [4]. Problems of these type have been the subject of immense interest in the past few years.

The necessary and sufficient conditions for generalized minimax programming were first developed by Schmitendorf [5]. Tanimoto [6] applied these optimality conditions to define a dual problem and derived duality theorems. Bector and Bhatia [7] relaxed the convexity assumptions in the sufficient optimality condition in [5] and also employed the optimality conditions to construct several dual models which involve pseudo-convex and quasi-convex functions, and derived weak and strong duality theorems. Yadav and Mukhrjee [8] established the optimality conditions to construct the two dual problems and derived duality theorems for differentiable fractional minimax programming. Chandra and Kumar [9] pointed out that the formulation of Yadav and Mukhrjee [8] has some omissions and inconsistencies and they constructed two modified dual problems and proved duality theorems for differentiable fractional minimax programming.

Lai et al. [10] established necessary and sufficient optimality conditions for non-differentiable minimax fractional problem with generalized convexity and applied these optimality conditions to construct a parametric dual model and also discussed duality theorems. Lai and Lee [11] obtained duality theorems for two parameter-free dual models of nondifferentiable minimax fractional problem involving generalized convexity assumptions.

Convexity plays an important role in deriving sufficient conditions and duality for nonlinear programming problems. Hanson [12] introduced the concept of invexity and established Karush-Kuhn-Tucker type sufficient optimality conditions for nonlinear programming problems. These functions were named invex by Craven [13]. Generalized invexity and duality for multiobjective programming problems are discussed in [14], and inseparable Hilbert spaces are studied by Soleimani-damaneh [15]. Soleimani-damaneh [16] provides a family of linear infinite problems or linear semi-infinite problems to characterize the optimality of nonlinear optimization problems. Recently, Antczak [17] proved optimality conditions for a class of generalized fractional minimax programming problems involving B-(p, r)-invexity functions and established duality theorems for various duality models.

In this article, we are motivated by Lai et al. [10], Lai and Lee [11], and Antczak [17] to discuss sufficient optimality conditions and duality theorems for a nondifferentiable minimax fractional programming problem with B-(p, r)-invexity. This article is organized as follows: In Section 2, we give some preliminaries. An example which is B-(1, 1)-invex but not (p, r)-invex is exemplified. We also illustrate another example which (-1, 1)-invex but convex. In Section 3, we establish the sufficient optimality conditions. Duality results are presented in Section 4.

2 Notations and prelominaries

Definition 1. Let f : X R (where X Rn) be differentiable function, and let p, r be arbitrary real numbers. Then f is said to be (p, r)-invex (strictly (p, r)-invex) with respect to η at u X on X if there exists a function η : X × X Rn such that, for all x X, the inequalities

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M1">View MathML</a>

hold.

Definition 2 [17]. The differentiable function f : X R (where X Rn) is said to be (strictly) B-(p, r)-invex with respect to η and b at u X on X if there exists a function η : X × X Rn and a function b : X × X R+ such that, for all x X, the following inequalities

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M2">View MathML</a>

hold. f is said to be (strictly) B-(p, r)-invex with respect to η and b on X if it is B-(p, r)-invex with respect to same η and b at each u X on X.

Remark 1 [17]. It should be pointed out that the exponentials appearing on the right-hand sides of the inequalities above are understood to be taken componentwise and 1 = (1, 1, ..., 1) ∈ Rn.

Example 1. Let X = [8.75, 9.15] ⊂ R. Consider the function f : X R defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M3">View MathML</a>

Let η : X × X R be given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M4">View MathML</a>

To prove that f is (-1, 1)-invex, we have to show that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M5">View MathML</a>

Now, consider

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M6">View MathML</a>

as can be seen form Figure 1.

thumbnailFigure 1. φ = sin2x + sin 2u(e-12(1+u) - 1) - sin2u.

Hence, f is (-1, 1)-invex.

Further, for x = 8.8 and u = 9.1, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M7">View MathML</a>

Thus f is not convex function on X.

Example 2. Let X = [0.25, 0.45] ⊂ R. Consider the function f : X R defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M8">View MathML</a>

Let η : X × X R and b : X × X R+ be given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M9">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M10">View MathML</a>

respectively.

The function f defined above is B-(1, 1)-invex as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M11">View MathML</a>

as can be seen from Figure 2.

However, it is not (p, r) invex for all p, r ∈ (-1017, 1017) as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M12">View MathML</a>

(for x = 0.4 and u = 0.42)

< 0 as can be seen from Figure 3.

Hence f is B-(1, 1)-invex but not (p, r)-invex.

In this article, we consider the following nondifferentiable minimax fractional programming problem:

(FP)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M13">View MathML</a>

where Y is a compact subset of Rm, l(., .): Rn × Rm R, m(., .): Rn × Rm R, are C1 functions on Rn × Rm and g(.): Rn Rp is C1 function on Rn. D and E are n × n positive semidefinite matrices.

Let S = {x X : g(x) ≤ 0} denote the set of all feasible solutions of (FP).

Any point x S is called the feasible point of (FP). For each (x, y) ∈ Rn × Rm, we define

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M14">View MathML</a>

such that for each (x, y) ∈ S × Y,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M15">View MathML</a>

For each x S, we define

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M16">View MathML</a>

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M17">View MathML</a>

with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M18">View MathML</a>

Since l and m are continuously differentiable and Y is compact in Rm, it follows that for each x* ∈ S, Y (x*) ≠ ∅, and for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M19">View MathML</a>, we have a positive constant

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M20">View MathML</a>

2.1 Generalized Schwartz inequality

Let A be a positive-semidefinite matrix of order n. Then, for all, x, w Rn,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M21">View MathML</a>

(1)

Equality holds if for some λ ≥ 0,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M22">View MathML</a>

Evidently, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M23">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M24">View MathML</a>

If the functions l, g, and m in problem (FP) are continuously differentiable with respect to x Rn, then Lai et al. [10] derived the following necessary conditions for optimality of (FP).

Theorem 1 (Necessary conditions). If x* is a solution of (FP) satisfying x*TDx* >0, x*TEx* > 0, and ∇gh(x*), h H(x*) are linearly independent, then there exist <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M25">View MathML</a>, ko R+, w, v Rn and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M26">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M27">View MathML</a>

(2)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M28">View MathML</a>

(3)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M29">View MathML</a>

(4)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M30">View MathML</a>

(5)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M31">View MathML</a>

(6)

Remark 2. All the theorems in this article will be proved only in the case when p ≠ 0, r ≠ 0. The proofs in the other cases are easier than in this one. It follows from the form of inequalities which are given in Definition 2. Moreover, without limiting the generality considerations, we shall assume that r > 0.

3 Sufficient conditions

Under smooth conditions, say, convexity and generalized convexity as well as differentiability, optimality conditions for these problems have been studied in the past few years. The intrinsic presence of nonsmoothness (the necessity to deal with nondifferentiable functions, sets with nonsmooth boundaries, and set-valued mappings) is one of the most characteristic features of modern variational analysis (see [18,19]). Recently, nonsmooth optimizations have been studied by some authors [20-23]. The optimality conditions for approximate solutions in multiobjective optimization problems have been studied by Gao et al. [24] and for nondifferentiable multiobjective case by Kim et al. [25]. Now, we prove the sufficient condition for optimality of (FP) under the assumptions of B-(p, r)-invexity.

Theorem 2 (Sufficient condition). Let x* be a feasible solution of (FP) and there exist a positive integer s, 1 ≤ s n + 1, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M32">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M33">View MathML</a>, ko R+, w, v Rn and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M34">View MathML</a> satisfying the relations (2)-(6). Assume that

(i) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M35">View MathML</a> is B-(p, r)-invex at x* on S with respect to η and b satisfying b(x, x*) > 0 for all x S,

(ii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M36">View MathML</a> is Bg-(p, r)-invex at x* on S with respect to the same function η, and with respect to the function bg, not necessarily, equal to b.

Then x* is an optimal solution of (FP).

Proof. Suppose to the contrary that x* is not an optimal solution of (FP). Then there exists an <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M37">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M38">View MathML</a>

We note that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M39">View MathML</a>

for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M40">View MathML</a>, i = 1, 2, ..., s and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M41">View MathML</a>

Thus, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M42">View MathML</a>

It follows that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M43">View MathML</a>

(7)

From (1), (3), (5), (6) and (7), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M44">View MathML</a>

It follows that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M45">View MathML</a>

(8)

As <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M46">View MathML</a> is B-(p, r)-invex at x* on S with respect to η and b, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M47">View MathML</a>

holds for all x S, and so for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M48">View MathML</a>. Using (8) and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M49">View MathML</a> together with the inequality above, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M50">View MathML</a>

(9)

From the feasibility of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M51">View MathML</a> together with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M52">View MathML</a>, h H, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M53">View MathML</a>

(10)

By Bg-(p, r)-invexity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M54">View MathML</a> at x* on S with respect to the same function η, and with respect to the function bg, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M55">View MathML</a>

Since bg(x, x*) ≥ 0 for all x S then by (4) and (10), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M56">View MathML</a>

(11)

By adding the inequalities (9) and (11), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M57">View MathML</a>

which contradicts (2). Hence the result.   □

4 Duality results

In this section, we consider the following dual to (FP):

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M58">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M59">View MathML</a> denotes the set of all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M60">View MathML</a> satisfying

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M61">View MathML</a>

(12)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M62">View MathML</a>

(13)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M63">View MathML</a>

(14)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M64">View MathML</a>

(15)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M65">View MathML</a>

(16)

If, for a triplet <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M66">View MathML</a>, the set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M67">View MathML</a>, then we define the supremum over it to be -∞. For convenience, we let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M68">View MathML</a>

Let SFD denote a set of all feasible solutions for problem (FD). Moreover, let S1 denote

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M69">View MathML</a>

Now we derive the following weak, strong, and strict converse duality theorems.

Theorem 3 (Weak duality). Let x be a feasible solution of (P) and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M70">View MathML</a> be a feasible of (FD). Let

(i) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M71">View MathML</a> is B-(p, r)-invex at a on S S1 with respect to η and b satisfying b(x, a) > 0,

(ii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M72">View MathML</a> is Bg-(p, r)-invex at a on S S1 with respect to the same function η and with respect to the function bg, not necessarily, equal to b.

Then,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M73">View MathML</a>

(17)

Proof. Suppose to the contrary that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M74">View MathML</a>

Then, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M75">View MathML</a>

It follows from (5) that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M76">View MathML</a>

(18)

with at least one strict inequality, since t = (t1, t2, ..., ts) ≠ 0.

From (1), (13), (16) and (18), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M77">View MathML</a>

Hence

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M78">View MathML</a>

(19)

Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M79">View MathML</a> is B-(p, r)-invex at a on S S1 with respect to η and b, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M80">View MathML</a>

From (19) and b(x, a) > 0 together with the inequality above, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M81">View MathML</a>

(20)

Using the feasibility of x together with μh ≥ 0, h H, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M82">View MathML</a>

(21)

From hypothesis (ii), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M83">View MathML</a>

As bg(x, a) ≥ 0 then by (14) and (21), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M84">View MathML</a>

(22)

Thus, by (20) and (22), we obtain the inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M85">View MathML</a>

which contradicts (12). Hence (17) holds.   □

Theorem 4 (Strong duality). Let x* be an optimal solution of (FP) and ∇gh(x*), h H(x*) is linearly independent. Then there exist <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M86">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M87">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M88">View MathML</a> is a feasible solution of (FD). Further, if the hypotheses of weak duality theorem are satisfied for all feasible solutions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M89">View MathML</a> of (FD), then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M90">View MathML</a> is an optimal solution of (FD), and the two objectives have the same optimal values.

Proof. If x* be an optimal solution of (FP) and ∇gh(x*), h H(x*) is linearly independent, then by Theorem 1, there exist <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M91">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M92">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M93">View MathML</a> is feasible for (FD) and problems (FP) and (FD) have the same objective values and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M94">View MathML</a>

The optimality of this feasible solution for (FD) thus follows from Theorem 3.   □

Theorem 5 (Strict converse duality). Let x* and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M95">View MathML</a> be the optimal solutions of (FP) and (FD), respectively, and ∇gh(x*), h H(x*) is linearly independent. Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M96">View MathML</a> is strictly B-(p, r)-invex at a on S S1 with respect to η and b satisfying b(x, a) > 0 for all x S. Furthermore, assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M97">View MathML</a> is Bg-(p, r)-invex at a on S S1 with respect to the same function η and with respect to the function bg, but not necessarily, equal to the function b. Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M98">View MathML</a>, that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M99">View MathML</a> is an optimal point in (FP) and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M100">View MathML</a>

Proof. We shall assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M101">View MathML</a> and reach a contradiction. From the strong duality theorem (Theorem 4), it follows that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M102">View MathML</a>

(23)

By feasibility of x* together with μh ≥ 0, h H, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M103">View MathML</a>

(24)

By assumption, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M104">View MathML</a> is Bg-(p, r)-invex at a on S S1 with respect to η and with respect to the bg. Then, by Definition 2, there exists a function bg such that bg(x, a) ≥ 0 for all x S and a S1. Hence by (14) and (24),

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M105">View MathML</a>

Then, from Definition 2, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M106">View MathML</a>

(25)

Therefore, by (25), we obtain the inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M107">View MathML</a>

As <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M108">View MathML</a> is strictly B-(p, r)-invex with respect to η and b at <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M109">View MathML</a> on S S1. Then, by the Definition of strictly B-(p, r)-invexity and from above inequality, it follows that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M110">View MathML</a>

From the hypothesis <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M111">View MathML</a>, and the above inequality, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M112">View MathML</a>

Therefore, by (13),

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M113">View MathML</a>

Since ti ≥ 0, i = 1, 2, ..., s, therefore there exists i* such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M114">View MathML</a>

Hence, we obtain the following inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/75/mathml/M115">View MathML</a>

which contradicts (23). Hence the results.   □

5 Concluding remarks

It is not clear that whether duality in nondifferentiable minimax fractional programming with B-(p, r)-invexity can be further extended to second-order case.

6 Competing interests

The authors declare that they have no competing interests.

7 Authors' contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Acknowledgements

Izhar Ahmad thanks the King Fahd University of Petroleum and Minerals for the support under the Fast Track Project no. FT100023. Ravi P. Agarwal gratefully acknowledges the support provided by the King Fahd University of Petroleum and Minerals to carry out this research. The authors wish to thank the referees for their several valuable suggestions which have considerably improved the presentation of this article.

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